What does Genesis 16:5 mean?

The plan to "help" God fulfill His promises by giving Hagar to Abram was Sarai's idea in the first place. A decade before (Genesis 16:3), God had once again promised to make Abram a great nation. This specifically included giving Abram a biological son (Genesis 15:4). For all of this time, Abram has chosen not to speed up God's timetable by marrying other wives. He has been waiting for God to fulfill the promise through Sarai, who was barren. Sarai thought the time had come to follow the custom of the day and marry her own servant to Abram to produce an heir. By that same custom, any heir would belong to Sarai as the "owner" of Hagar, her slave girl.

What Sarai seems to have failed to anticipate was Hagar growing contemptuous of her. Perhaps with good reason, Hagar would not relish the idea of giving up her child to Sarai and remaining the servant. She was Abram's wife, too, after all. Also, in that culture a woman's fertility was seen as a sign of her worth. So, Hagar's ability to conceive immediately for Abram, when Sarai had been childless for decades, could have been a source of pride or contempt for Hagar. Regardless of the reasons, or how Hagar might have acted, this contempt was not lost on Sarai.

Wounded by Hagar's attitude, Sarai complains to Abram. Her speech suggests she held him responsible for what had happened. In truth, though the plan was Sarai's idea (Genesis 16:2), Abram was responsible. He approved of the plan, he allowed it to happen, and he followed through with it. It was his responsibility to declare whether or not Sarai retained her authority over Hagar.

It's interesting to note that Sarai invokes the Lord's name in her angry plea to Abram. She calls on God to be the ultimate judge in this matter, going over Abram's head, in a sense. That was enough for Abram, as the next verse will reveal.